In evaluating composed functions, there are multiple methods to approach the problem, each with its own advantages. A composed function is formed by combining two functions, where the output of one function becomes the input of another. For instance, if we have two functions, f(x) = x² and g(x) = x - 1, we can find the composition f(g(x)) by substituting g(x) into f(x).

To illustrate, let's evaluate f(g(3)). First, we find the composition f(g(x)) by substituting g(x) into f(x):

f(g(x)) = f(x - 1) = (x - 1)²

Next, we simplify this expression:

(x - 1)² = (x - 1)(x - 1) = x² - 2x + 1

Now, to evaluate f(g(3)), we substitute x = 3 into the simplified expression:

f(g(3)) = 3² - 2(3) + 1 = 9 - 6 + 1 = 4

Thus, f(g(3)) = 4.

Alternatively, a shortcut method can be employed. Instead of finding the entire composition first, we can evaluate the inside function directly. For g(3), we compute:

g(3) = 3 - 1 = 2

Then, we substitute this result into f(x):

f(g(3)) = f(2) = 2² = 4

This method is often quicker, but it is essential to note that it may not always be applicable. In some cases, you will be required to find the composition f(g(x)) before evaluating it at a specific number. Therefore, while the shortcut can save time, understanding the full composition is crucial for certain problems.